SEQUENCE AND SERIES
INTRODUCTION
Sequences and series are fundamental concepts in mathematics that have wide applications in
various fields such as biology and finance. An understanding of these concepts not only strengthen
mathematical reasoning, but also provides insights into real world phenomena that are
characterized by consistent patterns and trends in our daily life
SPECIFIC INTRODUCTION
To gain an understand of this concept on a more deeper level, it is crucial to understand what these
two terms : sequences and series are, and more specifically how they work.
SEQUENCE : A sequence is an ordered list of number or objects which follow the same
pattern or set of rules.
Example:
2, 4, 6, 8, 10, ... (The sequence of even numbers)
1, 3, 5, 7, 9, ... (The sequence of odd numbers)
1, 1/2, 1/3, 1/4, 1/5, ... (The sequence of reciprocal of natural numbers)
SERIES : A series is a sum of the terms of a sequence. It is formed by adding up all the terms
in a sequence.
Example:
2 + 4 + 6 + 8 + 10 + ... (The series of even numbers)
1 + 3 + 5 + 7 + 9 + ... (The series of odd numbers)
1 + 1/2 + 1/3 + 1/4 + 1/5 + ... (The series of reciprocal of natural numbers)
WHATS A PROGRESSION?
A progression is a series of numbers following a specific pattern or rule. For example
3,6,9,12… is a progression as there is a pattern where every number here is obtained by
adding the digit 3 to its previous number.
But not all progressions are the same, there are different types such as arithmetic
progression, geometric progressions, etc.
Previously I had explained what progressions where, now I will be explaining some of its
types:
Arithmetic progression (AP): An arithmetic progression is a series of number in which the
difference between any two consecutive numbers are constant. The common difference
(constant) is added to each term to get the next term.
For example:
3, 6, 9, 12, 15... (Common difference = 3)
25, 20, 15, 10, 5…. (Common difference = -5)
Geometric progression (GP): A geometric progression is a sequence of numbers in which
each term after the first is found by multiplying the previous term by a fixed, non-zero
number called the common ratio (r).
For example:
2, 6, 18, 54, 162 ... (Common ratio = 3)
81, 27, 9, 3, 1 ... (Common ratio = 1/3)
DERIVATION OF THE FORMULAS
Common difference = d
Nth term = aₙ
Sum of the first n terms = Sₙ
Formula to find the nth term (aₙ) of an AP:
aₙ = a₁ + (n - 1) * d
Example: In the AP 3, 6, 9, 12, 15, ..., to find the 5th term (a₅), where a₁ = 3 and d = 3:
a₅ = 3 + (5 - 1) * 3 = 3 + 4 * 3 = 3 + 12 = 15
Formula to find the sum (Sₙ) of the first n terms of an AP:
Sₙ = (n/2) * [2a₁ + (n - 1) * d]
Example: In the AP 3, 6, 9, 12, 15, ..., to find the sum of the first 5 terms (S₅), where a₁ = 3
and d = 3:
S₅ = (5/2) * [23 + (5 - 1) * 3] = (5/2) * [6 + 43] = (5/2) * [6 + 12] = (5/2) * 18 = 45
Formula to find the nth term (aₙ) of a GP:
aₙ = a₁ * r^(n - 1)
Example: In the GP 2, 6, 18, 54, 162, ..., to find the 5th term (a₅), where a₁ = 2 and r = 3:
a₅ = 2 * 3^(5 - 1) = 2 * 3^4 = 2 * 81 = 162
Formula to find the sum (Sₙ) of the first n terms of a GP:
Sₙ = a₁ * (1 - r^n) / (1 - r)
Example: In the GP 2, 6, 18, 54, 162, ..., to find the sum of the first 5 terms (S₅), where a₁ = 2
and r = 3:
S₅ = 2 * (1 - 3^5) / (1 - 3) = 2 * (1 - 243) / (1 - 3) = 2 * (-242) / (-2) = 242
